C-Fiber Recovery Cycle Supernormality Depends on Ion Concentration and Ion Channel Permeability

Abstract

Following each action potential, C-fiber nociceptors undergo cyclical changes in excitability, including a period of superexcitability, before recovering their basal excitability state. The increase in superexcitability during this recovery cycle depends upon their immediate firing history of the axon, but also determines the instantaneous firing frequency that encodes pain intensity. To explore the mechanistic underpinnings of the recovery cycle phenomenon a biophysical model of a C-fiber has been developed. The model represents the spatial extent of the axon including its passive properties as well as ion channels and the Na/K-ATPase ion pump. Ionic concentrations were represented inside and outside the membrane. The model was able to replicate the typical transitions in excitability from subnormal to supernormal observed empirically following a conducted action potential. In the model, supernormality depended on the degree of conduction slowing which in turn depends upon the frequency of stimulation, in accordance with experimental findings. In particular, we show that activity-dependent conduction slowing is produced by the accumulation of intraaxonal sodium. We further show that the supernormal phase results from a reduced potassium current K_dr as a result of accumulation of periaxonal potassium in concert with a reduced influx of sodium through Na_v1.7 relative to Na_v1.8 current. This theoretical prediction was supported by data from an in vitro preparation of small rat dorsal root ganglion somata showing a reduction in the magnitude of tetrodotoxin-sensitive relative to tetrodotoxin -resistant whole cell current. Furthermore, our studies provide support for the role of depolarization in supernormality, as previously suggested, but we suggest that the basic mechanism depends on changes in ionic concentrations inside and outside the axon. The understanding of the mechanisms underlying repetitive discharges in recovery cycles may provide insight into mechanisms of spontaneous activity, which recently has been shown to correlate to a perceived level of pain.

Introduction

Our ability to detect and respond to potentially and overtly damaging stimuli is subserved by select subclasses of somatosensory C-neurons. The cell bodies of C-neurons reside in spinal ganglia just outside the spinal cord and each neuron projects a thin unmyelinated C-fiber axon of up to 1 meter in length to innervate peripheral tissues such as skin, muscle, and mucosa. Direct recordings from human C-fibers (1, 2, 3) show that this geometry, in particular the small intraaxonal volume, renders the propagation of each action potential (AP) highly dependent upon the immediate firing history of the axon. The axonal conduction velocity of APs generated at low frequency (ca. 2 Hz) decreases and can be halved during prolonged bouts of activity in a process referred to as activity-dependent slowing. In contrast, APs generated at high frequency (10–100 Hz) can speed up during axonal propagation, potentially increasing the interpulse frequency up to fivefold (1). Of clinical relevance is the observation that activity-dependent changes in axonal conduction are most prominent in mechanically insensitive or silent nociceptors. Silent nociceptors comprise ~30% of all C-fiber axons in humans, but their activation thresholds are typically too high for them to contribute to acute pain signaling. However, the threshold for activation of silent nociceptors can change, specifically a lowering of their threshold following inflammation and nerve injury and become active. In this way, silent nociceptors are particularly important for mechanical hyperalgesia (4, 5), axon reflex erythema (6), central sensitization (7), and probably spontaneous activity and thus ongoing pain in chronic pain states (8, 9, 10, 11, 12). Of particular importance is the tendency of silent nociceptors to show a phase of superexcitability ~10–100 ms after each AP, a feature that promotes high-frequency discharges.

Activity-dependent changes in axonal excitability are typically separated into short- and longer-term effects. Long-term effects are those occurring in the period beginning 300 ms after each AP. In this period a reduction in axonal conduction velocity is typically observed and accordingly long-term changes in excitability are referred to as activity-dependent slowing (ADS). When stimulated at relatively low (0.125 to 2 Hz) frequencies silent nociceptors show considerably more ADS than all other C-fiber classes (1, 2, 3). In humans, injection of nerve growth factor into the skin leads to an increase in the perceived intensity of electrically evoked pain (13, 14) and this change correlates with reduced ADS in silent nociceptors in the pig (15) and humans (B. Namer and O. Obreja, unpublished data). In experimental animal models of diabetic neuropathy, ADS is reduced in peripheral C-nociceptors, allowing nociceptive axons to follow higher AP discharge frequencies over longer periods (16). Recently, a correlation between reduced ADS in C-nociceptors and the incidence of spontaneous activity in nociceptors was described for patients with painful polyneuropathy (11).

The short-term effects associated with AP activity in C-fiber axons occur over an ~300 ms period following the passage of each AP. The characteristic series of axonal excitability changes seen in this period is termed the recovery cycle. The recovery cycle comprises an initial phase of reduced excitability followed by a facultative phase of enhanced excitability that subsequently reverses again to subexcitability. When measured experimentally as changes in electrical threshold the terms subexcitable-superexcitable-subexcitable are used, whereas for methods recording changes in axonal conduction latency, the terms subnormal-supernormal-subnormal are used. Propagation latency here refers to the time delay between stimulation at a peripheral site to the arrival of the AP at a proximal recording site. Typically two pulses are applied and the difference in latency of the second AP relative to the first is measured (1, 2, 3, 17). The processes underlying the cyclical changes in excitability have been extensively studied (1, 2, 3, 9, 18, 19, 20). The initial period of subexcitability is attributed to the inactivation of voltage-gated sodium channels. The facultative period of superexcitability correlates with a depolarizing after potential recorded directly from motor nerve terminals in the lizard and rat (21). The subsequent period of late subexcitability is attributed to hyperpolarization resulting from the electrogenic activity of the Na/K-ATPase (22). Recently, however, a prominent role of voltage-gated sodium channel inactivation and the accumulation of intracellular sodium have been shown to contribute to late subexcitability (20, 23, 24).

In humans, injections of nerve growth factor NGF led to increased pain ratings upon electrical skin stimulation in humans (13, 14) and reduced ADS of mechano-insensitive fibers in pig (15) and also in humans (B. Namer and O. Obreja, unpublished data). Of paramount clinical relevance are symptoms of spontaneous and ongoing pain, symptoms that are likely to originate from ectopic firing of peripheral nociceptors, including silent nociceptors (25, 26). Supernormality during the recovery cycle was found to be enhanced in diabetic patients (27) and can be suppressed by low-dose lidocaine (28). In light of these recent reports implicating a peripheral origin of spontaneous pain, we set out to examine the mechanisms determining excitability changes during the recovery cycle (1, 2, 3, 17) due to its potential importance for high-frequency discharge capacity. To this end, we have combined numerical simulations of AP conduction in an unmyelinated axon (29) with recordings of sodium currents in nociceptor cell bodies and in vivo recordings in humans. Our findings point to the importance of changes of ionic concentrations inside and outside the axon, and we will relate them to current literature in the discussion.

Materials and Methods

The model used in this study has been described in Tigerholm, Petersson et al. (29). Briefly, the model consists of two main geometrical and functional parts, a branch axon, and a parent axon (Fig. 1 A). The branch axon is 2 cm in length, has a diameter of 0.25 µm, and is set at a temperature similar to that found in superficial skin in humans (i.e., cold, 32°C). The parent axon is 10 cm long, has a diameter of 1 µm, and is set to a body temperature of 37°C. In comparing simulations to physiological in vivo cooling experiments, where a cooling plate was applied to the skin (see Microneurography below) we have assumed that the branch attains the temperature of the skin, whereas the parent maintains a body temperature. Ion concentrations of Na⁺ and K⁺ are represented within the axon as well as in a cylindrical shell (periaxonal space) and surrounding this as a constant extracellular concentration for each ion. The following ion channels are included: Na_v1.7, Na_v1.8, Na_v1.9, K_dr, K_A, K_M, K_Na, h (HCN). We also include the Na/K-ATPase ion pump.

Figure 1.

Compartmental C-fiber model and stimulation protocol. (A) C-fiber model consisting of a branch axon and a parent axon, connected via a cone. The branch and parent axons differ in length, diameter, and temperature (32 and 37°C for branch and parent, respectively), but have identical ion channel and pump densities as well as thickness of periaxonal space. The resting membrane potential for control conditions was set to -55mV. Stim = stimulation point, Rec = recording point on the C-fiber. (B) Stimulation protocol. RC1 has a variable ISI 10–250 ms. RC2 has a fixed ISI of 50 ms. In both protocols, repetition frequency is 2 Hz, pulse width is 5 ms, and pulse amplitude is 0.1 nA. To see this figure in color, go online.

In this work, we specifically analyzed the relative changes of Na_v1.7 vs. Na_v1.8. We therefore provide the full information on the respective currents, as described in Tigerholm, Petersson et al. (29). Q10_Na was 2.5 and E_Na was calculated from concentrations.

Na_v1.7: Values taken from Sheets et al. (30). g_Nav1.7 = 106.6 mS/cm².

$${\mathrm{I}}_{\mathrm{Nav}1.7}={\mathrm{g}}_{\mathrm{Nav}1.7}·{\mathrm{m}}^3\mathrm{hs}·\left({\mathrm{V}}_{\mathrm{m}}-{\mathrm{E}}_{\mathrm{Na}}\right)$$

$$\mathrm{m}=\mathrm{m}+\left(1-\exp \left(-\text{dt}/t_{\mathrm{m}}\right)\right)·\left({\mathrm{m}}_{\inf }-\text{m}\right)$$

$$a_{\mathrm{m}}=15.5/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}-5\right)/\left(-12.08\right)\right]\right\}$$

$${ß}_{\mathrm{m}}=35.2/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+72.7\right)/16.7\right]\right\}$$

$$t_{\mathrm{m}}=1/\left(a_{\mathrm{m}}+{ß}_{\mathrm{m}}\right)\times \mathrm{Q}{10}_{\mathrm{Na}}$$

$${\mathrm{m}}_{\inf }=a_{\mathrm{m}}/\left(a_{\mathrm{m}}+{ß}_{\mathrm{m}}\right)$$

$$\mathrm{h}=\mathrm{h}+\left(1-\exp \left(\mathrm{-dt}/t_{\mathrm{h}}\right)\right)·\left({\mathrm{h}}_{\inf }-\mathrm{h}\right)$$

$$a_{\mathrm{h}}=0.38685/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+122.35\right)/15.29\right]\right\}$$

$${ß}_{\mathrm{h}}=-0.00283+2.00283/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+5.5266\right)/\left(-12.70195\right)\right]\right\}$$

$$t_{\mathrm{h}}=1/\left(a_{\mathrm{h}}+{ß}_{\mathrm{h}}\right)\times \mathrm{Q}{10}_{\mathrm{Na}}$$

$${\mathrm{h}}_{\inf }=a_{\mathrm{h}}/\left(a_{\mathrm{h}}+{ß}_{\mathrm{h}}\right)$$

$$\mathrm{s}=\mathrm{s}+\left(1-\exp \left(\mathrm{-dt}/t_{\mathrm{s}}\right)\right)·\left({\mathrm{s}}_{\inf }\mathrm{-s}\right)$$

$$a_{\mathrm{s}}=0.00003+0.00092/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+93.9\right)/16.6\right]\right\}$$

$${ß}_{\mathrm{s}}=132.05-132.05/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}-384.9\right)/28.5\right]\right\}$$

$$t_{\mathrm{s}}=1/\left(a_{\mathrm{s}}+{ß}_{\mathrm{s}}\right)\times \mathrm{Q}{10}_{\mathrm{Na}}$$

$${\mathrm{s}}_{\inf }=a_{\mathrm{s}}/\left(a_{\mathrm{s}}+{ß}_{\mathrm{s}}\right)$$

Na_v1.8: Values taken from Sheets et al. (30) for activation (m³) and fast inactivation (h), whereas for slow (s) and ultraslow (u) inactivation, we used the equations and values from Maingret et al. (31), according to g = gbar·m³hsu. g_Nav1.8 = 242.7 mS/cm².

$${\mathrm{I}}_{\mathrm{Nav}1.8}={\mathrm{g}}_{\mathrm{Nav}1.8}·{\mathrm{m}}^3\mathrm{hsu}·\left({\mathrm{V}}_{\mathrm{m}}-{\mathrm{E}}_{\mathrm{Na}}\right)$$

$$\mathrm{m}=\mathrm{m}+\left(1-\exp \left(\mathrm{-dt}/t_{\mathrm{m}}\right)\right)·\left({\mathrm{m}}_{\inf }\mathrm{-m}\right)$$

$$a_{\mathrm{m}}=2.85-2.839/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}-1.159\right)/13.95\right]\right\}$$

$${ß}_{\mathrm{m}}=7.6205/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+46.463\right)/8.8289\right]\right\}$$

$$t_{\mathrm{m}}=1/\left({\mathrm{a}}_{\mathrm{m}}+{ß}_{\mathrm{m}}\right)\times \mathrm{Q}{10}_{\mathrm{Na}}$$

$${\mathrm{m}}_{\inf }=a_{\mathrm{m}}/\left(a_{\mathrm{m}}+{ß}_{\mathrm{m}}\right)$$

$$\mathrm{h}=\mathrm{h}+\left(1-\exp \left(\mathrm{-dt}/t_{\mathrm{h}}\right)\right)·\left({\mathrm{h}}_{\inf }-\mathrm{h}\right)$$

$$t_{\mathrm{h}}=1.218+42.043·\exp \left\{-\left[{\left({\mathrm{V}}_{\mathrm{m}}+38.1\right)}^2\right]/\left(2·{15.19}^2\right)\right\}\times \mathrm{Q}{10}_{\mathrm{Na}}$$

$${\mathrm{h}}_{\inf }=1/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+32.2\right)/4\right]\right\}$$

$$\mathrm{ds}/\mathrm{dt}=\left({\mathrm{s}}_{\inf }-\mathrm{s}\right)/t_{\mathrm{s}}$$

$${\mathrm{s}}_{\inf }=1/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+45\right)/8\right]\right\}$$

$$t_{\mathrm{s}}=1/\left(a_{\mathrm{s}}+{ß}_{\mathrm{s}}\right)\times \mathrm{Q}{10}_{\mathrm{Na}}$$

$$a_{\mathrm{s}}=0.001·5.4203/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+79.816\right)/16.269\right]\right\}$$

$${ß}_{\mathrm{s}}=0.001·5.0757/\left\{1+\exp \left[-\left({\mathrm{V}}_{\mathrm{m}}+15.968\right)/11.542\right]\right\}$$

$$\mathrm{du}/\mathrm{dt}=\left({\mathrm{u}}_{\inf }-\mathrm{u}\right)/t_{\mathrm{u}}$$

$${\text{u}}_{\text{inf}}=1/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+51\right)/8\right]\right\}$$

$$t_{\mathrm{u}}=1/\left(a_{\mathrm{u}}+{ß}_{\mathrm{u}}\right)\times \mathrm{Q}{10}_{\mathrm{Na}}$$

$$a_{\mathrm{u}}=0.0002·2.0434/\left\{1+\exp \left[\left({\mathrm{V}}_{\mathrm{m}}+67.499\right)/19.51\right]\right\}$$

The model replicates basic physiological characteristics such as spike width, conduction velocity, as well as activity-dependent slowing and recovery cycles. With regard to recovery cycles (RCs), the following criteria were used during construction of the model. The magnitude of supernormality should increase with increasing repetition frequency (2, 3), increasing repetition frequency of the double pulses (3), and a preconditioning pulse should not result in a time-dependent change in supernormality (3).

Stimulation protocols used

To study activity-dependent changes of AP conduction, we used the following stimulation protocols (schematically shown in Fig. 1 B). A current injection (5 ms, 0.1 nA) was applied at the distal end of the branch axon, leading to an AP that propagated to the end of the parent section. Latency was determined as the time between the injection of current at the branch terminal and the arrival of the AP at the proximal endpoint of the parent axon (Fig. 2). To assess the RCs and supernormal phase we used varying interstimulus intervals (ISI), similar to Weidner et al. (1) and Bostock et al. (3), (Fig. 2). In one protocol (RC1), we applied pulses at a repetition frequency of 2 Hz. After an initial train of prepulses, test pulses were added with varying interspike intervals (ISI). In a second protocol (RC2), pulse pairs with fixed ISI were applied at a constant repetition frequency (2).

Figure 2.

Slowing/speeding for different ISIs. The figure shows the slowing/speeding induced by stimulations with the RC1 protocol. The stimulation frequency is 2 Hz and the ISI varies between 10–250 ms. (A) Relative changes in conduction latency in human C-fiber shown as a function of interstimulus interval for a sequence of double pulses. The unit was recorded from the peroneal nerve at the knee and classified as a silent nociceptor on account of its lack of firing in response to both 75g mechanical stimuli in its electrical receptive field and sympathetic provocation with mental arithmetic. Upon cooling the receptive field from 32°C to 12°C, the conduction latency of the unit increased by 15% from 596 to 686 ms (inset). Determined at a base frequency of 0.25 Hz, the recovery cycle at 12°C is subnormal across all interstimulus intervals, whereas at 32°C a period of supernormal conduction is evident for interstimulus intervals between 40 and 350 ms. (B) Simulated membrane potential for different ISIs. The upper graphs represent the membrane potential in the beginning of the branch axon and the lower graphs the end of the parent axon. L = latency difference between first and second pulse. (C) Simulated application of RC1 protocol and effect of cooling. The difference between the latency of the second pulse and the first pulse is plotted versus the interspike interval, revealing slowing/supernormality for different ISIs. Simulations were performed at lower temperatures affecting the branch compartment of the model C-fiber, 22 (green), 12 (blue), and 2°C (cyan) and compared to the control temperature in our simulations of 32°C (red). To see this figure in color, go online.

Microneurography

The method of microneurography has been described in detail elsewhere (32). The recording electrode was inserted into the common peroneal nerve at the level of the fibular head. When the needle was inserted into a fascicle containing C-fibers, neuronal activity characteristic for unmyelinated C-fibers could be induced by scratching the skin on the dorsum of the foot. Innervation territories of individual C-fibers were then located with transcutaneous electrical stimulation with a pointed electrode (1–20 mA, 0.5 ms). C-fibers were identified by their low conduction velocity (<2 m/s), which was assessed from the latency of an electrically evoked AP after a rest period of at least 2 min. A pair of thin needles (0.15 mm diameter) was intracutaneously inserted into the innervation territory in a spot with low electrical threshold. Through these needles the C-fibers under observation were continuously stimulated at a low repetition rate (0.25 Hz) via a constant current stimulator (Digitimer DS7, Digitimer, Hertfordshire, UK). In response to repetitive electrical stimulation at low frequency, APs of individual C-fibers can be registered at constant latency at the recording electrode.

An increase in the usually stable conduction latency of peripheral C-fibers is observed (marking) when the stimulation frequency at the skin is increased or after the afferent fiber has been additionally activated, e.g., by natural heat or mechanical stimuli or by spontaneous activity (32). Marking is due to ADS of conduction in C-fibers, e.g., conduction of an AP renders the axonal membrane of afferent C-fibers less excitable for tens of seconds and thus slows down conduction velocity of subsequent APs.

During repetitive electrical stimulation at increasing frequencies (0.125, 0.25, 0.5 Hz) we determined activity-dependent conduction latency changes: after a rest period of at least 2 min; 20 electrical stimuli were applied intracutaneously at 0.125 Hz, immediately followed by trains of 20 pulses at 0.25 Hz and 30 pulses at 0.5 Hz (Fig. 2 A, inset). Changes in latency were calculated relative to the initial latency (i.e., that immediately following the 120 s rest period). If possible, we performed a second stimulation protocol that consisted of a train of pulses at 2 Hz for 3 min (17) and which was applied after another rest period of 2 min. Classification of fibers (mechano-insensitive, mechanosensitive, any sympathetic fibers) was based on characteristic slowing patterns and responses (marking) or lack thereof during mechanical stimulation or sympathetic provocation (see Fig. 2 A, inset) (17, 33, 34).

Whole-cell patch-clamp recordings

Dorsal root ganglions (DRGs) were isolated from rat pups (2–7 days-old) as reported previously (35) in accordance with ethical guidelines established by a German animal protection law, approved by the animal protection committee of Regierung Mittelfranken, Germany. Whole-cell patch-clamp recordings were performed from cultured DRG neurons after 1 or 2 days following dissociation using an EPC 10 amplifier at room temperature, 22 ± 1°C (HEKA electronics, Lamprecht, Germany). Room temperature was chosen as recordings at higher temperature on dissociated neurons are technically challenging, especially when the bath solution needs to be exchanged and recordings therefore need to last for a longer time period. Furthermore, the quality of the recording is more likely to vary over recording time at higher temperatures, adding variability to the subtractions of the current traces (I(TTXresistant) = I(total) – I(TTXsensitive), see below). Signals were filtered at 10 kHz and sampled at 50 kHz. The bath solution contained (in mM) 140 NaCl, 3 KCl, 1 CaCl₂, 1 MgCl₂, 10 Hepes, 20 TEA-Cl, 0.1 CdCl₂, and 5 d-glucose (pH 7.38). Glass pipettes (tip resistance 1.5–2 MO) were filled with internal solution containing (in mM) 140 CsF, 10 NaCl, 10 Hepes, 1 EGTA, 5 TEA-Cl, and 5 d-glucose (pH 7.3). The resulting liquid junction potential of 8.8 mV was not corrected for. All chemicals were purchased from Sigma (Germany) or Merck (Germany). Capacitive transients were cancelled and series resistances (<5 MO) were compensated by at least 55%. Small (<25 µm) DRG neurons were selected and held at -57 mV (for protocol -57mV and hyperpol) or -67 mV (for protocols -67mV and depol, see Fig. 6 B). The voltage protocols comprised two depolarization steps to 10 mV (for 3 ms, separated by 50 ms), revealing two fast sodium inward current peaks, I1 and I2. During the interpulse interval, cells were held either at -57 mV (protocols -57mV and depol) or -67 mV (protocols hyperpol and -67mV). To distinguish between tetrodotoxin (TTX) sensitive and resistant sodium currents, cells were perfused with 500 nM TTX (Biotrend, Switzerland) via a gravity-driven perfusion system and the remaining currents were subtracted from the native sodium current before TTX application. Results for TTX-sensitive (TTXs) and TTX-resistant (TTXr) currents are given as relative current amplitudes I2/I1. Data were analyzed using the FitMaster software (HEKA electronics) and the Igor Pro 5.2 software (Wavemetrics).

Figure 6.

Changes of Na_v1.7 and Na_v1.8 currents during double pulse voltage clamp. (A) The AP ADP increases over repetitive stimulations. The change is larger in the branch (open circles) than in the parent (×) and is larger for shorter ISIs. The figure shows the result from repeated stimulations at 2 Hz. Inset shows ISIs tested. (B) Four double pulse voltage protocols were used to determine current decline of Na_v currents in small diameter rat DRG neurons in whole-cell patch clamp experiments. In each protocol, two pulses to +10 mV (I1 and I2) were separated by 50 ms at varying interpulse potentials, as indicated. (C) Example whole-cell sodium current (traces in black) recordings using the -57 mV voltage protocol (top panel) in small diameter rat DRG neurons. After addition of 500 nM TTX to the external solution the TTXr component was recorded (blue traces). The TTXs component, (red traces) was revealed by subtraction of the traces recorded in the presence and absence of TTX. (D) Mean change in peak inward current I2 relative to I1 recorded with the voltage protocols shown in (B). Bars (left) labeled with TTXs (N = 17–21) and TTXr (N = 17–18) represent mean values +-SE from patch clamp recordings, whereas those labeled Na_v1.7 and Na_v1.8 (right) are simulation results at room temperature (RT) of experiments, 22°C and at regular body temperature of 37°C, respectively. Estimated experimental liquid junction potential of +8 mV was applied to the voltage-clamp simulation. To see this figure in color, go online.

Results

Excitability changes during RCs depend on accumulation of intracellular Na⁺

The recovery cycle comprises an initial phase of reduced conduction velocity of the second pulse in the doublet followed by a facultative phase of enhanced velocity that subsequently reverses to subexcitability again. We have examined the mechanisms potentially contributing to the excitability changes induced by RCs in unmyelinated axons using variable interstimulus intervals, ISIs, (RC1 (1, 3)) or fixed ISIs (RC2 (2)), schematically shown in Fig. 1 B, and investigated the contribution of different ionic conductances to phases of slowing or speeding. As we will uncover, most manipulations will lead to a reduction of ionic currents and supernormality will occur when currents reducing excitability are reduced more than currents enhancing excitability.

Simulation of empirical changes in axonal excitability during the recovery cycle and the effect of temperature

The changes in axonal conduction velocity that occur in the initial 500 ms or so following the passage of an AP have been extensively documented in people (1, 2, 3). An initial phase of reduced conduction velocity is followed by a period of supernormality that subsequently reverses to a second period of subnormality. The timing, magnitude and incidence of these three phases varies according to the axonal subtype. In particular, the extent and incidence of the supernormal phase (SNP) is dependent upon the repetition rate of stimulation. Recordings were made from single unmyelinated axons innervating human skin (Fig. 2 A). Initiating APs with electrical stimulation on the foot, the effects of conduction over a length of axon up to the peroneal recording site at the knee, show that human silent nociceptor (Fig. 2 A) can exhibit supernormality at a skin temperature of 32°. Previously undocumented however is the effect of temperature on the recovery cycle in human axons. Cooling the site of electrical stimulation in the skin to 12°C with a circular contact thermode of 10 mm diameter changes the shape of the recovery cycle such that the supernormal phase no longer manifests and the axon is always subnormal. This observation not only indicates that changes in axonal excitability are very much dominated by the terminal region of axons, where they are thinnest, but also provides direct empirical evidence for the role of temperature in scaling the recovery cycle. Specifically, with warming the processes underlying supernormality become more prominent. An illustrative example of a recovery cycle for a silent nociceptor recorded from the peroneal nerve is presented in Fig. 2 A. Consistent with previous reports, supernormality is evident at 0.25 Hz in the 30–70 ms interval (Fig. 2 A). Cooling the terminal portion of the axon slowed the absolute conduction latency (inset, Fig. 2 A) and resulted in a pronounced shift to subnormality throughout the recovery cycle (Fig. 2 A).

Simulations performed using paired stimulus pulses at variable ISIs (protocol RC1, Fig. 2, B and C), adequately emulated the overall trend of the change in latency of the second pulse relative to the first pulse. In particular, subnormality was observed at short ISI (<25 ms), supernormality was facultative and occurred at intermediate ISIs (peak around 40 ms; experimental range of peak 20–250 ms) and the second phase of subnormality extended over ISI values beyond ca. 250 ms. An increase in the magnitude of the supernormal phase was also observed with increasing stimulation (29), consistent with experimental observations (3). In addition, the simulation adequately reproduced the effects of altering temperature in the terminal region (Fig. 2 C) shifting the recovery cycle toward subnormality as observed in direct recordings from people (Fig. 2 A).

Effects of activity-dependent conditioning on supernormality at a fixed interstimulus interval (RC2)

To elucidate potential contributors to the transition from subnormality to supernormality, simulations were performed to determine relative changes in the conduction velocity of two APs initiated at a fixed interval (50 ms; protocol RC2). Consistent with the experimental findings of Weidner et al. (2), a change from subnormality to supernormality was observed only after an initial level of slowing had been achieved with repetitive activation (in Fig. 3 A after 5 double pulses at 2Hz). Consistent with the hypothesis regarding the effect of repetitive stimulation on axonal membrane potential (22) the simulated axon hyperpolarizes progressively during repetitive stimulations (Fig. 3 B) and this in turn increases the likelihood of a postspike depolarizing after potential (21). However, here and in previous simulations (29), a prominent and progressive increase in the concentration of intracellular sodium was also apparent during repetitive stimulation (Fig. 3 C).

Figure 3.

Induced supernormal phase due to accumulation of intracellular sodium. The figure shows the supernormal phase during the RC2 stimulation protocol. The stimulation frequency is 2 Hz and the ISI of a double pulse input is fixed to 50 ms. (A) Under control conditions (blue line) the supernormal phase is building up. When the concentrations of sodium and potassium are clamped to the initial value (black line) no SNP can be observed. Initial membrane potential (Vm) was -55mV. (B) The axon hyperpolarizes over repetitive stimulations. The figure shows the result from repeated stimulations at 2 Hz. Vm of parent and branch were sampled just before the following AP. (C) Intracellular sodium accumulates over repetitive stimulations. The figure shows the result from repeated stimulations with RC2 at 2 Hz. Concentrations (Conc) in mM. Fluctuations in extracellular potassium follow APs and visually fuses at this time base. (D) SNP for one double pulse when the initial value of the intraaxonal sodium concentration was systematically changed. (E) SNP for one double pulse when the resting membrane potential was hyperpolarized and either clamping ion concentrations to the resting concentration value at -55mV (black) or allowing ion concentrations to change as in control simulations (blue). To see this figure in color, go online.

Independent effects of changes in ionic concentration and membrane potential on supernormality

The increase in concentration of intraaxonal sodium (Fig. 3 C) that parallels the development of hyperpolarization (Fig. 3 B) during repetitive stimulation could readily reflect the causal link between these two parameters via the Na⁺/K⁺-ATPase (21, 22). To examine these two parameters in isolation, in silico techniques were used to clamp either the intraaxonal sodium concentration or membrane potential. Supernormality was assessed by monitoring the conduction latency with protocol RC2 (ISI = 50 ms). Concentration was varied independently of membrane potential to examine their respective role in supernormal axonal conduction. In the first set of simulations, intraaxonal sodium concentration was varied while keeping the resting membrane potential at -55 mV. With the resting membrane potential fixed, supernormal conduction developed in parallel with increasing intracellular sodium concentration, beginning at ~15 mM (Fig. 3 D).

In contrast, with the intraaxonal sodium concentration held constant at 11.4 mM hyperpolarizing the membrane potential increased subnormality, i.e., hyperpolarization alone reduced the likelihood of supernormality (conc. fixed, Fig. 3 E). However, if the concentration of sodium is allowed to change, then hyperpolarization of the membrane leads to the emergence of supernormality (conc. free, Fig. 3 E) consistent with in vitro data (36). These results suggest that 1) an increase in the concentration of intraaxonal sodium is necessary and sufficient for supernormal conduction independent of membrane potential and 2) that hyperpolarization alone cannot account for supernormality.

Differential contribution of individual Na_v isoforms to supernormality

The requirement of an elevation in intraaxonal sodium concentration for the emergence of supernormality during repetitive activity (Fig. 3 D) prompted an examination of the role of specific voltage-dependent sodium channel (Na_v) subtypes in mediating increases in intraaxonal sodium. Taking advantage of the ability to clamp the Na⁺ reversal potential for individual Na_V isoforms in silico, the reversal potential for Na_V1.8 was first held at its initial value of 69 mV. In comparison to the control condition, (Fig. 4 A, blue) this led to a pronounced increase in the development of the supernormal phase (Fig. 4 A, red). In contrast, clamping the Na reversal potential for Na_v1.7 at 69 mV resulted in a cumulative increase in subnormality (Fig. 4 A, green). Thus, keeping the reversal potential for Na_v1.7 constant and thereby allowing Na_v1.7 to maintain its amplitude opposes development of a supernormal phase and instead leads to increased slowing. From this observation, we predicted that for the second AP of a pair, the likelihood of supernormality might increase as the relative contribution of Na_v1.8-mediated current increased.

Figure 4.

Role of Na_v subtypes in SNP. (A) Latency differences are shown during the RC2 stimulation protocol when the reversal potential is selectively clamped for Na_v1.7 (green line) or Na_v 1.8 (red line) and under control condition with variable reversal potential (blue line) The stimulation frequency is 2 Hz and the ISI is fixed to 50 ms. (B) The ratio between Na_v1.7 and Na_v1.8 for the first pulse pair is shown at their respective stimulation time P1 = first pulse, P2 = second pulse in doublet. (C) The ratio between Na_v1.7 and Na_v1.8 for the last (200th) pulse pair is shown at their respective stimulation time. For (D)–(E) the black curve represents the ratio between Na_v1.7 and Na_v1.8 for the 200th pulse pair. Note that arrival times of P1 differ between conditions due to different CV. (D) Changing intraaxonal sodium concentration to 50 mM, corresponding to the value obtained after 200 pulses (red). (E) Changing the resting membrane potential to -60 mV, corresponding to the value obtained after 200 pulses (blue). (F) Changing both resting membrane potential and the intraaxonal sodium concentration as in (D) and (E) (green), comparable to the situation in (C). To see this figure in color, go online.

Fig. 4, B–F, shows the effect of varying intraaxonal sodium concentration and membrane potential on the relative current amplitude of Na_v1.7 relative to Na_v1.8 for two APs generated 50 ms apart. To do this, the magnitude of Na_v1.7-mediated to Na_v1.8-mediated current was determined in the early phase of the AP when Na_v1.7 currents are active. In the absence of the preceding activity, the Na_v1.7 current amplitude for the first pulse in the doublet was only ~70% of the Na_v1.8 current (Fig. 4 B). However, for the second AP that initiated 50 ms after the first, the relative contribution of Na_v1.8 current increased, i.e., the Na_V1.7/Na_V1.8 ratio fell to ~40%. This more pronounced decline of Na_v1.7 current compared to Na_v1.8 is due to Na_V1.7’s slower repriming kinetics (37). When challenged with repetitive double pulse activity at 2 Hz, the Na_v1.7/Na_v1.8-ratio for the leading pulse remained basically unchanged at 70% (Fig. 4 C). However, the shift in favor of Na_v1.8 in the trailing AP became more pronounced and the relative Na_V1.7 current fell to around 25% of its Na_V1.8 counterpart (Fig. 4 C). This suggests that in axons expressing both Na_v1.7 and Na_v1.8, the likelihood of supernormality during repetitive activity increase as the relative Na_v1.7-mediated current falls.

Because an increase in intraaxonal sodium is necessary for supernormality, the effect of sodium concentration on the ratio of Na_v1.7/Na_v1.8-mediated current was examined for concentrations present at the first and the last of a 2 Hz series of 200 stimulations with RC2 (ISI 50 ms), 11.4 and 50 mM respectively. Raising the intracellular sodium concentration to 50 mM, results in a more pronounced reduction in the ratio of Na_v1.7 to Na_v1.8 mediated current after 200 pulses at 2 Hz. (Fig. 4 D). An increase in the intraaxonal Na⁺ concentration thus favors Na_v1.8-mediated current relative to Na_v1.7. This originates from a reduction in total sodium currents leading to a widening of the AP (29), in turn leading to increased Na_v1.7 inactivation. In contrast, the relative change from pulse 1 to pulse 200 of Na_v1.7/Na_v1.8-mediated current remains similar to control if the Na concentration is clamped to 11.4 mM and only the membrane potential is hyperpolarized to -60 mV (Fig. 4 E). However, hyperpolarization substantially enhances the Na_v1.7/Na_v1.8-ratio for the leading pulse from 0.65 up to ~2.5, whereas for the trailing AP the ratio is ~2/3 of that during the first pulse (Fig. 4 E). The combined effect of increasing the intracellular sodium concentration and hyperpolarizing the membrane potential alters the Na_v1.7/Na_v1.8 ratio to levels achieved after 200 pulses of the RC2 protocol (compare Fig. 4 F to Fig. 4 C). Thus, the change in Na_v1.7/Na_v1.8-ratio can result from an increase in intraaxonal sodium concentration and membrane hyperpolarization. For the control condition shown in Fig. 4, B and C, we conclude that for the leading AP the more pronounced suppression of Na_v1.7 relative to Na_v1.8 from increased Na⁺ concentration is balanced by an enhancement of Na_v1.7 from the hyperpolarization. However, for the second AP in the doublet, the enhancing effect of hyperpolarization is less effective because Na_v1.7 availability is low due to its slow repriming. Thus, during repetitive stimulation, changes in membrane potential and Na⁺ concentration lead to a shift in favor of Na_v1.8 over Na_v1.7-mediated sodium current during the conducted AP. As shown in Fig. 4 A, enhanced currents through Na_v1.8 favor SNP, and thus our simulations offer a molecular explanation for SNP.

Accumulation of extracellular potassium promotes postspike supernormality

Based on our findings that changes of intracellular Na⁺ concentration as well as clamping the sodium reversal potential affected supernormality, we were interested in studying the possible involvement of K⁺ concentration and K⁺ reversal potential. In Fig. 3 A we have shown that an increase of intraaxonal sodium constitutes a necessary condition for generation of the SNP. There is however a second condition, which is unmasked by the reduction of Na⁺ current during repetitive spiking: changes of extracellular potassium ion concentrations. To dissociate changes in concentrations from membrane potential, we used our established protocol and studied a single double pulse during different conditions. As discussed previously, intracellular Na⁺ accumulates over repetitive stimulation (Fig. 3 C). We therefore compared simulations of a single double pulse with an ISI of 50 ms performed under control conditions (resting, Na⁺_i = 11.4 mM) with those under elevated intraaxonal Na⁺ (50 mM), corresponding to the condition after 200 pulse pairs at 2 Hz (Fig. 5). Consistent with results from the RC1 protocol presented in Fig. 2, initial slowing (Fig. 5 A, black line) is replaced by supernormality (Fig. 5 B, green line) at elevated intraaxonal sodium concentration.

Figure 5.

SNP depends on potassium ion concentration. (A) Resting intraaxonal sodium concentration (Na_in = 11.4 mM). Membrane potential (Vm, top) and potassium reversal potential (ErevK, middle) recorded in the parent axon. Propagation latency (bottom). Latency changes from the first to second pulse. ISI = 50 ms. Variable K⁺ reversal potential (black line), clamped K⁺ reversal potential (blue line). (B) Elevated sodium concentration (Na_in = 50 mM). Membrane potential (Vm, top) and potassium reversal potential (ErevK, middle) recorded in the parent axon. Propagation latency (bottom). Latency changes from the first to second pulse. ISI = 50 ms. Variable K⁺ reversal potential (green line), clamped K⁺ reversal potential (red line). (C) Latency differences measured by RCs following additional preconditioning pulses. In the graph, latency difference was computed between second and third pulses, (red lines and markers). Latency differences in the original RC1 protocol are included for comparison (blue lines and markers). To see this figure in color, go online.

The main potassium current during the AP repolarization in our model is carried by a delayed rectifier-type potassium current (IK_dr). In simulations performed with the potassium reversal potential clamped to its initial value, an increase of subnormal conduction was observed, both at control (11.4 mM; Fig. 5 A, blue line), and at elevated intraaxonal Na⁺ concentrations (50 mM; Fig. 5 B, red line). Thus, in addition to sodium concentration changes, potassium concentration can also contribute to RC changes of excitability. A fixed reversal potential would correspond to a large unrestricted extracellular space, which is not present under physiological conditions. If the periaxonal space is restricted in volume, as is the case for a nerve fiber in the body, the extracellular concentration of K⁺ increases in the wake of the first action potential such that the reversal potential for potassium is reduced at the time of initiation of the second action potential 50 ms later (Fig. 5 A, middle panel). The ensuing reduction in potassium currents lowers the spike threshold, increases the rate of upstroke, and speeds the second AP (Fig. 5 A, top panel). Moreover, consistent with observations in Fig. 4, increasing the intraaxonal Na⁺concentration further enhances supernormality (Fig. 5 B). An additional prepulse before the double pulses was suggested to affect periaxonal ion concentrations (3). To address this, we introduced an extra prepulse to test the hypothesis of ion accumulation. The simulation showed a decrease in the magnitude of the SNP (Fig. 5 C, red markers) consistent with the findings of Bostock et al. (3). However, in contrast to the interpretation made by Bostock et al. (3), our results were obtained in the presence of increasing potassium concentration.

Role of after depolarization and resting membrane potential in supernormality

The size of the peak of the afterdepolarization (ADP) between the recorded pulses did not change in our simulations, independent of whether SNP occurred or not (Fig. 5, top). This indicates that a postspike depolarization does not constitute a necessary condition for supernormality. This result thereby contrasts the argument that an ADP after the first pulse leads to the SNP (2, 3, 21).

Given the clear empirical evidence for a depolarizing afterpotential in the time window in which supernormality in unmyelinated axons appears (21), we investigated its potential influence on supernormality. During repetitive stimulation, we find an increase in the magnitude of the ADP, Fig. 6 A. This is in agreement with recordings from myelinated axons (21). In contrast to the empirical data of Barrett and Barrett (21), for the simulated axon the peak of the ADP increases by a maximum of 2.2 mV, whereas the absolute potential of the ADP decreases due to the 5 mV hyperpolarization of the resting membrane potential, as seen in Fig. 3 B. The transition from subnormal to supernormal phase observed during repetitions can therefore not be explained based on the ADP. Comparing the peak amplitude of ionic currents generated during the AP, we find that over 200 repetitions of paired pulses, amplitudes of Na_v1.7 and Na_v1.8 currents decline, as a result of accumulation of intracellular sodium ions and channel inactivation, as described previously. For the K_dr current, there is also a reduction due to a reduced potassium driving force caused by the hyperpolarization. Comparing sodium and potassium currents for the first AP in the doublet, sodium and potassium current decreases are approximately equal over 200 repetitions. However, all factors taken together, we find that over 200 repetitions, the peak current decrease between first and second pulse in the doublet is 2% larger for the K_dr potassium current than for the sum of sodium currents Na_v1.7 and Na_v1.8 (data not shown). Together with the concomitant shift in favor of Na_v1.8 over Na_v1.7, supernormality is obtained.

To sum up our findings so far, we have shown that supernormality depends on several factors. Accumulation of extracellular potassium after the first AP results in a reduction of K_dr for a second AP generated 50 ms later. This supports enhanced propagation velocity of the second AP. This change in potassium current magnitude is only effective on propagation velocity if the velocity is already reduced, as a result of, for instance, repeated stimulations (preexisting slowing). This long-term reduction in conduction velocity is influenced by the accumulation of intraaxonal sodium ions lowering the reversal potential of sodium currents. Finally, the differential effects by sodium channel subtypes results from the difference in the level of inactivation, which is moderate for Na_v1.8 but quite substantial for Na_v1.7 within the period of supernormality. The smaller reduction of Na⁺-currents and the larger reduction of the K⁺-current together lead to supernormality.

Effect of membrane potential on TTXs and TTXr current amplitude in response to paired depolarizing stimuli in rat DRG neurons

In our simulations, hyperpolarization leads to a relative enhancement of Na_v1.7 over Na_v1.8 (Fig. 4 E). We therefore examined the differential effect of membrane potential on Na_v1.7 and Na_v1.8. We hypothesized that hyperpolarization between the two APs would lead to a more complete recovery from inactivation for Na_v1.7 than for Na_v1.8. To this end to study the interplay between membrane voltage and ion channel kinetics, voltage clamp experiments were performed on acutely isolated small rat DRG neurons in vitro. Acute application of the sodium channel blocker TTX was used to isolate TTXs and TTXr Na_v currents (see Fig. 6 C) and the results were compared to corresponding simulation results in voltage clamp. Average current amplitudes for TTXs and TTXr current for the protocols are found in Table 1. For the model, simulations were made at both room temperature, 22°C and body temperature, 37°C.

Table 1.

Current amplitudes (mean, standard deviation, number of observations)

	TTXs	TTXs	TTXs	TTXr	TTXr	TTXr
I1 (nA)	I2 (nA)	I2/I1	I1 (nA)	I2 (nA)	I2/I1
-57 mV	-2.69	-2.37	0.916	-6.96	-6.72	0.929
2.11 (n = 17)	1.73 (n = 17)	0.117 (n = 17)	8.97 (n = 17)	8.85 (n = 17)	0.059 (n = 17)
hyperpol	-2.27	-2.34	1.08	-6.01	-5.91	0.980
2.11 (n = 20)	2.11 (n = 20)	0.194 (n = 20)	7.91 (n = 17)	7.80 (n = 17)	0.020 (n = 17)
-67 mV	-2.66	-2.02	0.847	-4.48	-4.37	0.965
2.73 (n = 21)	1.75 (n = 21)	0.140(n = 21)	3.39 (n = 18)	3.38 (n = 18)	0.025 (n = 18)
depol	-2.58	-1.52	0.749	-4.45	-4.12	0.886
2.64 (n = 21)	1.13(n = 21)	0.267(n = 21)	3.21 (n = 18)	3.21 (n = 18)	0.082 (n = 18)

TTXs (tentatively dominated by Na_v1.7) and TTXr (tentatively dominated by Na_v1.8) peak inward Na⁺-currents were determined in response to a pair of depolarizing voltage steps to +10 mV from a holding potential of either -57 mV or -67 mV. In the 50 ms period between the two voltage pulses the membrane potential was similarly held at either -57 mV or -67 mV (protocols are shown in Fig. 6 B). The interspike interval of 50 ms was chosen as it matches the approximate location of the largest supernormality (Fig. 2) and would therefore be appropriate to study mechanisms in supernormality. The effect of an intervening period of either depolarization or hyperpolarization on the relative magnitude of TTXs and TTXr Na⁺-current for paired pulses is shown in Fig. 6 D. The most prominent result is that only an intervening period of hyperpolarization can enhance the TTXs current relative to the TTXr current (protocol hyperpol, Fig. 6 D). When the potential during the ISI is either depolarized (protocol depol) or has the same value as the holding potential (protocols -57mV or -67mV), TTXs current was suppressed relative to its TTXr counterpart. This experimental result was replicated with the model (Fig. 6 D). Specifically, the relative magnitude of Na_v1.7/Na_v1.8-mediated current was smaller for the second pulse for all conditions other than for an intervening period of hyperpolarization. Our experiments and simulations in voltage clamp therefore are consistent with our findings in current clamp shown in Fig. 4.

Thus, using whole-cell patch-clamp recordings of Na⁺-currents, we find:

• 1)In three of the four protocols used, the size of the Na⁺-current at the second pulse is reduced compared to that at the first pulse. In all protocols, except protocol hyperpol for Na_v1.7, the Na_v current is smaller for the second pulse.
• 2)Reduction of TTXr currents at the second pulse is smaller than for TTXs currents (2–11%). However, the size of current reduction differs significantly between the protocols (protocol -57 is different from protocol hyperpol; protocol depol differs from protocol -57, hyperpol, and -67; one-way analysis of variance (p < 0.001) and Fisher least significant difference post-hoc test). Thus, the current modification is sensitive to the protocol used but the difference in TTXr current modification is modest.
• 3)For TTXs over all four protocols, we first note that changes span a larger range than for TTXr currents, from enhancement of 8% to suppression of 25%. Current amplitudes are significantly different between protocols (protocol -67 differs from hyperpol; protocol depol differs from -57, and hyperpol; protocol -57 differs from depol, one-way analysis of variance (p < 0.001) and Fisher least significant difference post-hoc test).
• 4)For the four protocols, current amplitudes between TTXs and TTXr currents differ significantly for the hyperpol protocol and the -67mV protocol (Student’s t-test): Whereas a holding potential of -67 mV results in a suppression of TTXs currents, a holding potential of -57 mV with a hyperpolarization during the ISI shows an enhancement of TTXs currents. It is also interesting to note that the protocol depol results in the strongest reduction of TTXs, and relatively smaller decrease in TTXr current, shifting Na_v currents in favor of TTXr currents. Furthermore, given the ISI interval of 50 ms and kinetics of Na_v1.8 inactivation, it is possible that the small Na_v1.8 reduction could be smaller or reversed for shorter depolarizations. Moreover, during SNP the potential right before the second APS is depolarized (Fig. 5, A and B), thus driving Na_v1.7 into inactivation and reducing the Na_v1.7/Na_v1.8 ratio, as shown in our simulations. We were able to mimic this effect with our experimental results, showing that a depolarization between the two pulses results in a much stronger decrease of TTXs currents compared to TTXr currents (Fig. 6 D).

As we have shown in Fig. 2 A, cooling to 12°C suppresses supernormality in human silent nociceptors. Of importance, our simulation readily replicates this effect of cooling (Fig. 2 C). In which case, currents recorded in DRG neurons in vitro at an intermediate temperature of 22°C, although likely to underestimate magnitudes at body temperature, are still to reflect the contribution of TTXs and TTXr currents relative to one another. Our computational modeling indicate that shifts may occur at higher temperatures (37°C, Fig. 6 D), in particular that for Na_v1.7, the second pulse may increase relative to the first for the hyperpolarizing protocol, whereas it may decrease for the depolarizing protocol (Fig. 6 D). Thus, our combined approach of intracellular recordings and computational simulations provide mechanistic insight into the relative behavior of Na_v isoforms over consecutive spikes.

Taken together, simulations and experiments suggest that a 50 ms afterhyperpolarization reduces and a 50 ms ADP enhances the relative contribution of Na_v1.8 over Na_v1.7-mediated current for the second AP in a 50 ms doublet. This supports the initial hypothesis based on the model prediction that an after hyperpolarization can rescue Na_v1.7 from inactivation and an ADP or maintained membrane potential will favor Na_v1.8, providing further support for the validity of the model studied.

Discussion

We have used simulations of repetitive activity in unmyelinated axons to examine the potential mechanistic underpinnings of changes in axonal excitability, known as the recovery cycle, that follow each conducted AP. In general, axons are subexcitable and conduct at subnormal velocities immediately after a single AP. Superimposed on this subnormality is a facultative period of supernormal conduction, during which a second AP of the same fiber is conducted faster than the preceding one. The magnitude of this supernormality increases with the level of preceding activity. The supernormal phase of axonal AP conduction has been difficult to examine experimentally. Our simulations indicate that supernormality results from a relative smaller decrease in the magnitude of sodium over potassium currents. Of importance, the dynamic changes in the recovery cycle seen during ongoing activity can be emulated by activity-dependent changes in the accumulation of intraaxonal sodium, leading to a general reduction in CV, and accumulation of periaxonal potassium. The latter accumulation results from influx during the first pulse of the doublet, reducing the driving force of potassium currents during the second pulse. More specifically, in our work, we show supernormality is a result of reduced K_dr current in concert with a reduced contribution of Na_v1.7 relative to Na_v1.8. The importance of these shifts of channel activity is contingent upon slowing generated by the accumulation of intraaxonal sodium. We show that the Na_v-subtypes contribute differently to SNP, with Na_v1.8 promoting and Na_v1.7 opposing its appearance. We also show that the two major sodium currents differ in how they respond to ADPs and therefore differ in how they contribute to recovery cycles. Na_v1.7 is more suppressed by the ADP due to its strong and fast inactivation at depolarized potentials. They also respond differently to the membrane hyperpolarization during repetitive stimulation. In this respect, Na_v1.7 recovers from inactivation to a larger degree than Na_v1.8. The final outcome (supernormality or enhanced subnormality) thus depends on an intricate interplay between ion concentrations, membrane potential, and Na_v-subtype activity. The use of computational modeling is particularly well suited to address questions like these.

Relations to experimental findings

Our results indicate that supernormality increases when the amplitude of Na_v1.7-mediated inward current is reduced relative to that of Na_v1.8. A role for TTXs Na⁺ currents in determining axonal recovery cycles in myelinated sensory and motor axons has been empirically verified in humans after ingestion of TTX in the form of puffer fish poisoning (38). In this group of four patients, an unknown but relatively high dose of TTX resulted in a reduction in compound AP amplitude and reduction in the magnitude of supernormality. This observation, with a change of supernormality at odds with our results, is restricted to A-fibers. Furthermore, as we have shown here, supernormality depends on a number of factors, including membrane potential, ion concentrations inside and outside the axon, and ion channel availability, and it is possible that one or several of these factors differ between the A-fibers and C-fibers studied in the TTX-poisoned patients and in the control conditions forming the basis of our model.

Based on our simulation results of the differential role of sodium channel subtypes (Fig. 4), we hypothesized that hyperpolarization between the two APs following a double pulse stimulation would lead to a more complete recovery from inactivation for Na_v1.7 than for Na_v1.8. Our in vitro recordings of Na_v-mediated current refine this hypothesis, indicating that a relative shift in magnitude of currents from different Na_v-isoforms could potentially underlie the emergence of supernormality (Fig. 6). Our results further suggest potassium currents are reduced in an activity-dependent manner. This is consistent with results in an unmyelinated nerve (39). Moreover, the reduction in K_dr activity that we find during the second pulse depends on accumulation of periaxonal potassium. This is consistent with experimental findings of elevated extracellular potassium with activity (40) and reduction of conduction efficacy due to potassium accumulation (41). Our findings of activity-dependent relations between ionic currents, driving force, and reversal potential based on quantitative modeling are consistent with suggestions put forward by Lüscher et al. (41).

RC mechanism

Several hypotheses concerning the supernormal phase during recovery cycles have been put forward. The earliest hypothesis suggested that the increase in the velocity during the supernormal phase would be due to an accumulation of extracellular potassium (21). Our results are supporting this hypothesis. Accumulation of potassium ions reduces its reversal potential and thereby the magnitude of the potassium currents. Thus, the balance between potassium and sodium currents is shifted and excitability is increased. This hypothesis was tested in a study by Bostock et al. (3). In the experiment, an extra pulse was added just before the double pulse. The authors assumed that if the accumulation of potassium contributes to the supernormal phase, supernormality should increase with the extra pulse. However, this was not found in the study. We tested the same stimulation protocol as Bostock et al. (3) in the model. Surprisingly, the result was consistent with the Bostock et al. (3) results (decrease of SNP), but we show that the increase of periaxonal K⁺ supports SNP nevertheless, albeit at a different timescale. This shows how problematic it is to draw conclusions from indirect measurements and the advantages of a computational model.

The most accepted hypothesis today proposes that passive properties of the membrane potential generate the ADP (2, 3, 9, 18). ADP has been measured in hyperpolarized myelinated axons (21) and is the foundation on which this hypothesis is built upon. More convincing evidence was presented in a study by Bowe et al. (42), where the ADP would lead to an increase in excitability with very similar dynamics as measured during recovery cycle protocol. Our model results are consistent with the interpretation that the ADP could be beneficial in the generation of the supernormal phase. It induces a larger inactivation in Na_v1.7 than in Na_v1.8. shifting the balance in favor of Na_v1.8, which according to our findings led to the shift from slowing to supernormality. Additionally, in fibers with an A-type potassium current contributing to the AP repolarization, an ADP could lead to channel inactivation and thereby further facilitate supernormality by reducing the K_A-current at the second pulse.

Furthermore, in the study of Moalem-Taylor et al. (36) current injections were used to depolarize the membrane and study the effect on the SNP. Their finding that the SNP increases with depolarization was interpreted as a support for the hypothesis of the ADP as the mechanism of the SNP. However, our study shows that membrane potential depolarization also leads to a change in intracellular ion concentration, and we further show that this concentration change is the determining factor. Our study further shows depolarization alone is not sufficient to produce a SNP. In our work, we show that the basic phenomena producing supernormality is due to a change in periaxonal potassium concentration reducing the driving force of potassium. This leads to a lowering of the spike threshold and an increasing rate of depolarization, both leading to an increased CV. These results are consistent with the experimental observations in a human KCNA1 loss-of-function mutation (43) where effects of reduced delayed rectifier K_v1.1 currents on peripheral nerve were studied.

Recovery cycles and pain

In light of the recent interest in the relationship between pain and high spontaneous activity (8, 9, 10, 11, 12), and the relationship between repetitive spiking and spontaneous activity (9, 44), we were interested in studying mechanisms of repetitive spike propagation. Velocity recovery cycles constitute a well-studied phenomena of spike repetition up to frequencies of 500 Hz (1, 2, 3, 17). Interestingly, in diabetic patients, recovery cycle supernormality was found to be enhanced (27). Understanding the mechanisms behind recovery cycles, and in particular supernormality, may therefore provide understanding of changes present in chronic pain. Our results point to an interplay between sodium and potassium channels, where the relative larger reduction of potassium currents over sodium currents constitute a prerequisite for supernormality. We further show that the process also involves a shift in relative contribution from Na_v1.7 to Na_v1.8. The study thereby highlights the complex interplay between ionic currents and aspects of excitability of the nerve.

Editor: Randall Rasmusson.